Is vector A divided by magnitude of a, is a scalar or a vector. I think it is a vector because the direction is not eliminated.

My other question is, using vectors explain why |a+b| is always less than |a|+|b|.

I think it is always smaller than |a|+|b| because it goes in a straight line, from point a to point c, and not a-to-b-to-c. For example, |a+b| of the 3,4,5 triangle is 5, while |a|+|b| is 7. This means you will have a less magnitude because you will be going in a straight line.